Part One: Moons of other planets.
The discovery of the Galilean moons of Jupiter in 1609-1610 proved that not everything had to orbit directly around one single object.
The discovery of the Galilean moons showed that objects could revolve around an object which revolved around another object. There was no way for all known objects to revolve directly around each other.
- The Sun, the Moon, and the planets revolved around Earth, and the 4 Galilean moons revolved around Jupiter, thus only indrectly orbiting the Earth.
- Earth, the Moon, and the planets orbited around the Sun, and the 4 Galilean moons revolved around Jupiter, thus only indirectly orbiting the Sun.
- The Sun, the Earth, the other planets, and the 4 Galilean moons, all revolved around Jupiter, and the Moon revolved around Earth, thus only indirectly orbiting Jupiter.
Decades later, Titan was discovered in 1654, and four other large moons of Saturn were discovered in 1671 to 1684.
So now there were three objects in the solar system which certainly had objects orbiting them, Earth, Jupiter, and Saturn, and there was a big astronomical question whether everything ultimately orbited the Earth or the Sun.
Part Two: Phases of the Planets.
When telescopes were first used to observe the planets starting in 1609-1610, the phases of the planets began to be observed, phases caused by the changing angles between the Sun, the planets, and Earth.
It became noticed that the phases of Mercury and Venus went all the way from thin crescents to 100 percent full, and that their angular diameters were larger when they were thin crescents and smaller when they were full. Half the times they appeared close to the Sun they appeared small and full, and half the times they appeared close to the son they appeared large and crescent shaped. That was consistent with them orbiting around the Sun at a distance less than the distance between Earth and the Sun.
But the phases of Jupiter and Saturn were different. They ranged from slightly more than half full to totally full. Jupiter and Saturn always appeared smaller and less full the closer they were to the direction of the Sun but were never less than half full. They never appeared larger and less than half full when they were close to the direction of the Sun. So they must have orbited around either the Earth or the Sun at distances greater than the distance between the Earth and the Sun.
Part Three: Parallax.
Diurnal parallax is a parallax that varies with rotation of the Earth or with difference of location on the Earth. The Moon and to a smaller extent the terrestrial planets or asteroids seen from different viewing positions on the Earth (at one given moment) can appear differently placed against the background of fixed stars.
The diurnal parallax has been used by John Flamsteed to measure the distance to Mars at its opposition and through that to estimate the astronomical unit and the size of the solar system.
john Flamsteed lived from 1646 to 1719.
After Copernicus proposed his heliocentric system, with the Earth in revolution around the Sun, it was possible to build a model of the whole Solar System without scale. To ascertain the scale, it is necessary only to measure one distance within the Solar System, e.g., the mean distance from the Earth to the Sun (now called an astronomical unit, or AU). When found by triangulation, this is referred to as the solar parallax, the difference in position of the Sun as seen from the Earth's centre and a point one Earth radius away, i. e., the angle subtended at the Sun by the Earth's mean radius. Knowing the solar parallax and the mean Earth radius allows one to calculate the AU, the first, small step on the long road of establishing the size and expansion age of the visible Universe.
A primitive way to determine the distance to the Sun in terms of the distance to the Moon was already proposed by Aristarchus of Samos in his book On the Sizes and Distances of the Sun and Moon. He noted that the Sun, Moon, and Earth form a right triangle (with the right angle at the Moon) at the moment of first or last quarter moon. He then estimated that the Moon–Earth–Sun angle was 87°. Using correct geometry but inaccurate observational data, Aristarchus concluded that the Sun was slightly less than 20 times farther away than the Moon. The true value of this angle is close to 89° 50', and the Sun is actually about 390 times farther away. He pointed out that the Moon and Sun have nearly equal apparent angular sizes and therefore their diameters must be in proportion to their distances from Earth. He thus concluded that the Sun was around 20 times larger than the Moon; this conclusion, although incorrect, follows logically from his incorrect data. It does suggest that the Sun is clearly larger than the Earth, which could be taken to support the heliocentric model.
Although Aristarchus' results were incorrect due to observational errors, they were based on correct geometric principles of parallax, and became the basis for estimates of the size of the Solar System for almost 2000 years, until the transit of Venus was correctly observed in 1761 and 1769. This method was proposed by Edmond Halley in 1716, although he did not live to see the results. The use of Venus transits was less successful than had been hoped due to the black drop effect, but the resulting estimate, 153 million kilometers, is just 2% above the currently accepted value, 149.6 million kilometers.
Much later, the Solar System was "scaled" using the parallax of asteroids, some of which, such as Eros, pass much closer to Earth than Venus. In a favourable opposition, Eros can approach the Earth to within 22 million kilometres. Both the opposition of 1901 and that of 1930/1931 were used for this purpose, the calculations of the latter determination being completed by Astronomer Royal Sir Harold Spencer Jones.
Also radar reflections, both off Venus (1958) and off asteroids, like Icarus, have been used for solar parallax determination. Today, use of spacecraft telemetry links has solved this old problem. The currently accepted value of solar parallax is 8".794 143.
Part Four: Stellar Parallax.
The heliocentric theory implies that as the Earth orbits the Sun, the angles to stars should change. Because of the vast distances to the stars, astronomers failed to detect the parallaxes and distances to the stars for centuries.
Advances in instruments and techniques led to the first successfull measurements of stellar parallax in the 1830s, when the parallaxs and distances of Alpha centauri, 61 Cygni, and Vega were first measured. Alpha Centauri is the closest star system to the Sun, 61 Cygni is the 15th, and there should be a few hundred star systems closer - and easier to measure their parallaxes - than Vega.
Part Five: Conclusion.
Any amateur astronomer who has telescopes, instruments, and observing techniques as good as those of professional astronomers of 17th and 18th centuries, or the 19th century up to the 1830s, can duplicate those observations which give stronger and stronger evidence that the Earth orbits the Sun instead of the Sun orbiting the Earth.